Most of us are familiar with the Math Subject
Classification
(MSC),
a coded index attempting to classify all mathematical
research areas by topic. The MSC, devloped jointly by the Math Reviews and Zentralblatt, is used by most journals and many grant institutions,
such as the US National Science Foundation, as a way of
grouping mathematical work into topic categories. The MSC
codes were recently updated from the year 2000 codes to the
current 2010 Mathematics Subject
Classification. These codes are organized hierarchically, first
dividing into broad research areas, then into sections and
finally into more specific research categories.

Question. How well do these codes describe the natural divisions of
research in mathematics? Could they be improved in some
way? How should they be revised?

Most of us, when submitting a research article for
publication, have to decide on the most appropriate codes
for that particular work. My own experience is that usually
there there is a natural code or two codes that fit very
well, which aptly describe the research topic of the
article. Sometimes I use two or more codes in a situation
where the work doesn't really fit well into either of them
alone, so that it isn't really a primary/secondary
classification for me, but rather a classification into the
union of two categories. Increasingly, however, I find
myself stymied by the classification scheme, frustrated in
my newest projects that perhaps four or five subcategories
are involved, with none of them truly apt, except for the
unhelpful "None of the above, but in this section"
category. In such cases, I feel that the MSC has failed
me.

I recognize that this may simply mean that I sometimes
favor offbeat topics, and so perhaps this is my problem
rather than the MSC's problem. Or perhaps my problem is
that I would like my research to be categorized by the
bottom level of the hierarchy, but I should be content just
with using the middle level of the hierarchy.

At the same time, I recognize that the mathematical
community has a specific interest in encouraging research
that crosses the boundaries between established areas,
perhaps cross-pollinating or unifying them or at least
transferring methods and techniques from one area to
another. In time, therefore, we expect subject
classification boundaries to migrate or split in various
ways. Indeed, perhaps some of the most valuable
mathematical work tends to destroy the old classification
scheme for precisely this kind of reason. Presumably, this
is part of the reason why the MSC is somewhat regularly
updated (every ten years I think). So I suspect that there may be many people who share my frustration.

How would you revise the MSC?

Let's have standard community-wiki rules; please provide
just one group of changes per post.

Perhaps also someone could briefly explain the relationship between the MSC codes and the Math arXiv codes.
–
Joel David HamkinsJun 16 '10 at 2:26

It's perhaps worth mentioning that some universities use these codes or their own variants for their performamce/research metrics. For instance, I was once asked as a PhD student to choose from a list of (somewhat ill-fitting) codes for data being collected on the university's research strengths.
–
Q.Q.J.Jun 16 '10 at 13:13

8 Answers
8

I would add a top-level subject code roughly corresponding to the arXiv math.QA. Currently my field (quantum groups, knot invariants, TQFT, monoidal categories, etc.) is listed under 16T, 17B, 20G42, 58B32, 81R50, 57R56, 81T, 57M, 56L37, and 18D. But there's no top-level classification that's appropriate, I end up having to describe myself as 17 "Nonassociative rings and algebras" which is terribly misleading. Math.QA is perfect, and the MSC just doesn't have anything like it.

I think MSC is a historical anachronism, often useful for bureaucratic purposes, but mathematically indefensible. For one, it is built as a tree with some weak "for xyz see ..." connectors, while a better form would be some kind of poset of subareas. Also, the reason arXiv seems better is because 1) it was invented later and 2) it has only large areas. If arXiv has sub-areas, 15 years later it would be just as bad.

Some years ago I was distressed by how Wikipedia treated the subject as well and completley rewrote/restructured the Combinatorics article, which is still more or less in the way I have made it. Based on that, let me comment only on MSC 05 (Combinatorics). Here is what we have:

Now, 05A is a fine category as long as you don't try to look at its sub-cats. For example, 05A40 is "Umbral calculus". Quick, show of hands for those who think this sub-area is comparable with 05A05 which is "Permutations, words, matrices". But let's not go there - more trouble is coming up.

05B. This would be a coherent choice if this was "codes, block designs and sphere packing" or something like that. However, by looking at some sub-areas you quickly realize this category has no structure at all. Essentially, anything can be "configuration" in the opinion of the MSC authors. For example, 05B includes "Matroids" (large area) and "Polyominoes" (not an area at all, sort of similar to "unit cubes"), which are and should be viewed as distant parts of Geometric Combinatorics. It also includes "Matrices" (hello?) and "Difference sets" (huh?). The most beautifully titled category is 05B99 "None of the above, but in this section", and given that 05B has no common theme, anything goes, I guess...

05C. In contrast with 05B this is a very clear and coherent category. It is also very popular and on a permanent quest for independence (the suggestion being that it becomes 07 which for whatever historical reason is missing in MSC).

05D. Shouldn't this be "Extremal and Probabilistic Combinatorics"? So if a paper proves a new result in Ramsey theory using probabilistic method, is it 05D40 (Probabilistic methods) or 05D10 (Ramsey theory)? Anyway, this is a clear category which should be linked to 60C (Combinatorial probability), which itself needs to be renamed "Discrete Probability".

05E. This would a clear category if the areas were more connected. As it stands, 05E30 (Association schemes, strongly regular graphs) should really go into 05B. Similarly, 05E40 is an important sub-area and should really go into MSC 13, or perhaps replaced by a better named sub-cat (every part of Algebra now has "combinatorial aspects" - should we list them all in 05E?), or even become a separate part of 05 (or of both 05 and 13, if MSC becomes a poset). Most strikingly, another important area 05E45 (Combinatorial aspects of simplicial complexes) should be taken outside of 05E (what exactly is algebraic about it?) and made into a separate part of 05 titled "Topological Combinatorics".

Ugh... In summary, this mess has to be largely redone. New parts of 05 need to be created (we have 21 letters F..Z remaining for "Geometric", "Topological", "Analytic", "Arithmetic", etc. - see Wikipedia page). What is left in each category would then be more coherent and searching for such cats in MathSciNet would actually make sense.

Thanks for this thoughtful answer, Igor. I agree with your general assessment, and similar issues arise in section 03.
–
Joel David HamkinsJun 17 '10 at 0:29

1

Speaking of Wikipedia, its data structure of "categories" (still hierarchical, but no longer a tree, and even with an occasional directed closed loop) is superior for organizing material if the ease of searching is the goal.
–
Victor ProtsakJun 17 '10 at 1:09

@Victor: that's right. Another important WP feature is that it allows a quick and easy update when things change. Unfortunately, if you make changes to MSC too often that also diminishes the value of the classification system. Some kind of middle ground works best in this case.
–
Igor PakJun 17 '10 at 3:14

To be honest, I have never in my life paid any attention to this stuff (except for its sieving application when reading postdoc job applications). I read papers because of the title/abstract, or knowing what the content is about. I pick a couple when forced to by journals, and didn't think anyone actually cares. It's like the "Keyword" stuff I am forced to create (which again I never look at for papers I read). Are those actually used for something?

I agree that for NSF logistics and job applications it is useful. But the question is focused on its role in research papers, and that is where I don't get it, and is what my puzzlement is about. I never look for papers in my areas of interest by searching for MSC numbers, and I never look for papers by searching for a "Keyword". I am really baffled by the tradition of putting these things in research papers.

MSC was created in order to categorize or sort the papers being reviewed. As with many other things, it acquired life of its own...
–
Victor ProtsakJun 16 '10 at 4:32

@Victor: Oh, so its purpose is to make life easier for the folks at Math Reviews (e.g., to funnel the paper to the correct editor there)? And the Keywords help them to more easily pass the paper on to suitable reviewers? That makes a lot of sense. So it is like the story of US Social Security numbers.
–
BoyarskyJun 16 '10 at 5:06

1

There have been a small number of times when I've found keywords and the MSC useful. About 7 years ago, I decided I wanted to know about everything that was known about Witt vectors. It was pretty easy to find lots of new papers by doing MSC and keyword searches, and I found certain veins of knowledge, probably much earlier than I would have otherwise. More generally, the MSC and keywords are probably useful whenever you want to know what people know about a very specific topic that cuts across many different fields and which has an accepted name.
–
JBorgerJun 16 '10 at 6:17

1

@Boyarsky: It's unfair to the Math Reviews staff to characterize the classification that way. They do need a fairly stable and hierarchical classification scheme in order to assign papers to editors and then to contact appropriate reviewers. But the scheme is also needed by users to retrieve information over many decades. It's easy to pass suggestions along to MathSciNet for changes, but these occur slowly in collaboration with Zentralblatt and have to preserve the archival search framework. I've persuaded them over the years to refine the classification in areas of interest to me.
–
Jim HumphreysJun 16 '10 at 12:01

@Jim: Your description of its uses is consistent with what I was thinking; if the way I wrote made it sound flippant, that was not intentional (perhaps the phrase "make life easier" was not a good way to put it). I know how much effort they put into these things (and know some people who work at MR). The basic upshot is that I honestly had no idea (until Protsak mentioned it) that MR actually used these in their work, though it is obvious in hindsight. Now that I know this, I will be more attentive to it with my papers!
–
BoyarskyJun 16 '10 at 12:18

I suspect that any attempt to produce a hierarchal top-down style tree of mathematical subjects is bound to be problematic. You might make the majority of the people happy at the time you create the tree (like the arXiv now) but long-term it's likely to have the same problems that people see with the MSC classification.

The history of mathematics is that deep connections are found between fields that are at earlier times perceived as distant. So fields glom together. Similarly, fields drift apart (like math.GT and math.AT now, in the 50's there was just topology) for various reasons, evolution of techniques being one of them.

My initial guess would be that the best long-term solution would be to have an MSC classification that is as "flat" as possible. So that when you pick your MSC classification, it's like choosing flavours at an ice-cream shop.

My experience has been that the editors of Math Reviews pay attention to suggestions about revisions of the classification system. I'm not saying that they implement all the suggestions (especially because the suggestions sometimes contradict each other), but they do listen and, a while before the once-a-decade revision of the system, they ask some people (including me on some occasions) about both the proposed changes and any other ideas we might want to contribute. The revision process is non-trivial, partly because of the need (or at least desire) to coordinate with Zentralblatt, and partly because a major revision of any section has a big downside (the loss of backward compatibility) and must therefore have a big upside to make it worthwhile. But big revisions have happened, and they may well happen again. The editors also watch for classification areas that have gotten either unpleasantly small (so they might be merged into other areas) or unpleasantly large (so they might be split, if natural dividing lines can be found).

Based on my own experience for MSC2010 I can confirm that editors do listen to suggestion, even from people they did not ask. There was a public call for suggestions, and the process of revision (at least in part) was public as well. The webpage for this (I think so far it did not get mentioned) is still visible msc2010.org ; it was advertised when the procsess was current on MathSciNet and/or Zentralblatt MATH. My experience contributing to an (unsolicited) suggestion was very positive and in the end it was (in part) implemented.
–
quidAug 30 '12 at 18:06